Mathematics • Year 10 • Unit 2 • Lesson 18

Equations of Lines — Skill Drill

Build fluency with the three "find the equation" methods from Lesson 18: from gradient + y-intercept (direct y = mx + c), from gradient + one point (find c), and from two points (find m, then c). Also convert between y = mx + c and general form Ax + By + C = 0.

Build · I Do / We Do / You Do

1. I do — fully worked example

Find the equation of a line through two points.

Problem. Find the equation of the line through (−2, 5) and (4, −1) in y = mx + c form.

Step 1 — Compute the gradient.

m = (−1 − 5)/(4 − (−2)) = −6/6 = −1

Step 2 — Use the y = mx + c form with m = −1.

y = −x + c

Step 3 — Substitute one of the points to find c.

Using (−2, 5): 5 = −(−2) + c = 2 + c → c = 3

Step 4 — Write the equation and check with the other point.

y = −x + 3. Check (4, −1): −4 + 3 = −1 ✓

Answer: y = −x + 3.

Stuck? Revisit lesson § "From Two Points" — Worked Example 3.

2. We do — fill in the missing steps

Convert y = mx + c form to general form. Fill in each blank. 5 marks

Problem. Write y = (−3/4)x + 5 in general form Ax + By + C = 0 with integer coefficients and a positive A.

Step 1 — Multiply every term by ____ (the denominator) to clear the fraction:

____ y = ____ x + ____

Step 2 — Move every term to the LHS so the equation equals 0:

____ x + ____ y − ____ = 0

Step 3 — If A is negative, multiply through by −1 to make it positive:

Result: ____ x + ____ y − ____ = 0

Step 4 — Identify A, B, C: A = ____, B = ____, C = ____.

Step 5 — Verify by checking the y-intercept (set x = 0): ____ y − 20 = 0 → y = ____ ✓ (matches original).

Stuck? Revisit lesson § "General Form".

3. You do — independent practice

For each: write the equation in the form requested. Always verify by substituting one point.

Foundation — from m and y-intercept

3.1 Gradient 3, y-intercept −2. Write y = mx + c. 1 mark

3.2 Gradient −1/2, y-intercept 7. Write y = mx + c. 1 mark

3.3 Gradient 2, passes through (1, 5). Find c. 1 mark

3.4 Gradient −2 through (1, 5). Write y = mx + c. 1 mark

Standard — gradient + point, two points

3.5 Find the equation through (2, 3) and (4, 7). 2 marks

3.6 Find the equation through (−1, 2) and (3, −6). 2 marks

Extension — convert between forms

3.7 Convert y = (2/3)x − 4 to general form Ax + By + C = 0 with integer coefficients and positive A. 3 marks

3.8 Convert 5x − 2y + 8 = 0 to y = mx + c form and state the gradient and y-intercept. 3 marks

Stuck on 3.7? Multiply by 3 → 3y = 2x − 12 → 2x − 3y − 12 = 0.

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Answers — Do not peek before attempting

Section 2 — We do (y = (−3/4)x + 5)

Step 1: × by 4 → 4y = −3x + 20. Step 2: 3x + 4y − 20 = 0 (already has positive A). Step 3: A is already positive, so result unchanged: 3x + 4y − 20 = 0. Step 4: A = 3, B = 4, C = −20. Step 5: at x = 0 → 4y = 20 → y = 5 ✓.

3.1 — m = 3, c = −2

y = 3x − 2.

3.2 — m = −1/2, c = 7

y = −(1/2)x + 7.

3.3 — m = 2 through (1, 5)

5 = 2(1) + c → c = 3.

3.4 — m = −2 through (1, 5)

5 = −2(1) + c → c = 7. y = −2x + 7.

3.5 — (2, 3) and (4, 7)

m = (7 − 3)/(4 − 2) = 2. y = 2x + c. Using (2, 3): 3 = 4 + c → c = −1. y = 2x − 1. Check (4, 7): 8 − 1 = 7 ✓.

3.6 — (−1, 2) and (3, −6)

m = (−6 − 2)/(3 − (−1)) = −8/4 = −2. y = −2x + c. Using (−1, 2): 2 = 2 + c → c = 0. y = −2x. Check (3, −6): −6 = −2(3) ✓.

3.7 — y = (2/3)x − 4 → general form

× 3: 3y = 2x − 12. Rearrange: 2x − 3y − 12 = 0 (A = 2 already positive ✓).

3.8 — 5x − 2y + 8 = 0 → y = mx + c

−2y = −5x − 8 → y = (5/2)x + 4. Gradient m = 5/2; y-intercept c = 4.